Continuous Data Assimilation for the 2D B\'enard Convection through Velocity Measurements Alone

TL;DR

This paper introduces a novel continuous data assimilation algorithm for 2D Bénard convection that uniquely relies on velocity measurements alone, ensuring exponential convergence to the true solution without temperature data.

Contribution

It develops a new data assimilation method for Bénard convection using only velocity data, extending previous Navier-Stokes algorithms to thermal convection problems.

Findings

Algorithm guarantees exponential convergence under certain spatial resolution conditions.

Velocity-only data assimilation is sufficient for accurate reconstruction of the flow.

Conditions on data resolution ensure noise-free data leads to reliable convergence.

Abstract

An algorithm for continuous data assimilation for the two- dimensional B\'enard convection problem is introduced and analyzed. It is inspired by the data assimilation algorithm developed for the Navier-Stokes equations, which allows for the implementation of variety of observables: low Fourier modes, nodal values, finite volume averages, and finite elements. The novelty here is that the observed data is obtained for the velocity field alone; i.e. no temperature measurements are needed for this algorithm. We provide conditions on the spatial resolution of the observed data, under the assumption that the observed data is free of noise, which are sufficient to show that the solution of the algorithm approaches, at an exponential rate, the unique exact unknown solution of the B\'enard convection problem associated with the observed (finite dimensional projection of) velocity.